Print Page - Multi-dimensional complex plane equivalent ... finally?

An important property of the complex plane is the power property:
$i^1=i,\;i^2=-1,\;i^3=-i,\;i^4=1,\;\cdots$
If the real unit is specified as $j$ , then the power property would look like this:
$i^1=i,\;i^2=-j,\;i^3=-i,\;i^4=j,\;\cdots$
And I see no reason why the pattern could not be continued into higher dimensions:
3D: $i^1=i,\;i^2=-j,\;i^3=-k,\;i^4=-i,\;i^5=j,\;i^6=k,\;\cdots$
4D: $i^1=i,\;i^2=-j,\;i^3=-k,\;i^4=-l,\;i^5=-i,\;i^6=j,\;i^7=k,\;i^8=l,\;\cdots$
5D: $i^1=i,\;i^2=-j,\;i^3=-k,\;i^4=-l,\;i^5=-m,\;i^6=-i,\;i^7=j,\;i^8=k,\;i^9=l,\;i^{10}=m,\;\cdots$
etc. So in 3D, with the formula $\left\{ z_n=(z_{n-1})^2+k\;|\;z_0=2i+3j \right\}$ ,
$z=\{2i+3j,-9i-4j+13k,-250i-185j+242k,\cdots\}$
Just to be sure, for the last step, I calculated $-9i-4j+13k$ as $-9i-4i^5+13i^6$ and $9i^4+4i^2+13i^6$ and got the same answer.
So could a true 3D Mandelbrot or Julia fractal be made like this? Please reply.

-For the glory of God

P.S. I had to do a lot of the math for this post by hand. Do any of you know of a good program that can do this kind of thing for me?

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Fractal Software => 3D Fractal Generation => Topic started by: truth14ful on May 07, 2011, 05:27:30 PM